Phase Separation of Liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ Mixtures: Effect of Confinement

We study the phase separation of helium mixtures inside thin channels in contact with a bulk ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ bath at zero temperature. It is found that the substrate potential favors mixing near the walls. While in bulk, the transition takes place between a ${}_{}{}^4{}_{}{}^{}\mathrm{He}$-rich mixture with 6.6% of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$; in thin channels, the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$-rich phase is richer in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-rich phase contains a nonzero fraction of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. We anticipate that both phases can be superfluid, so that the superfluid line is detached from the coexistence region. The behavior of helium mixtures in porous aerogel is discussed in relation to the present findings.